A finite nonseparable
topological graph G in E2 is said to be a triangular map if all its finite faces are
triangular. Edges and vertices of G are external if they are incident with the infinite
face, otherwise they are internal. The maps considered are rooted by distinguishing
an external vertex and incident external edge. A polygon in the graph G of
such a map is Hamiltonian if it includes afl vertices of G. In this paper,
the average number of Hamiltonian polygons in a member of the class of
nonisomorphic rooted triangular maps with n internal and m + 3 external vertices is
determined. Asymptotic estimates are included for the results obtained. An
unexplained coincidence is shown between the number of Hamiltonian polygons in
rooted triangular maps and in their duals, rooted nonseparable trivalent
maps.
